statistics - topic 4 - sampling distributions

Question 1

what is the formula for a sample mean?

Answer 1

𝑋̅=1/𝑛 ∑𝑋

Question 2

what is the standard error of the mean of a sample?

Answer 2

𝜎_𝑋̅ =𝜎/√𝑛

Question 3

what is the formula for the variance of a non independent sample?

Answer 3

𝑉𝑎𝑟(𝑋̅ )=𝜎^2/𝑛
x (𝑁−𝑛)/(𝑁−1)

Question 4

what is the finite population correction factor?

Answer 4

(𝑁−𝑛)∕(𝑁−1)

Question 5

what is the finite population correction factor used for?

Answer 5

it accounts for the fact the individual sample members are not independent of each other and as N-n<N-1 then it will reduce the standard deviation

Question 6

if a population is normally distributed with a mean 𝜇 and standard deviation 𝜎 , then what is the formula for the sample mean and standard deviation?

Answer 6

𝐸[𝑋̅ ]=𝜇_𝑋̅ =𝜇 and 𝜎_𝑋̅ =𝜎/√𝑛

Question 7

why is the sample mean an unbiased estimator of 𝜇

Answer 7

because both distributions will have the same mean

Question 8

why is 𝜎_𝑋̅ not a unbiased estimator of 𝜎

Answer 8

this is due to the fact that The distribution of 𝑋̅ has a smaller standard deviation than 𝑋

Question 9

what does central limit theorem state?

Answer 9

As 𝑛 becomes large, the central limit theorem states that the distribution of
𝑍=(𝑋̅−𝜇_𝑋̅ )/𝜎_𝑋̅
approaches the standard normal distribution. ie even if the population is not normally distributed when sample size is large enough it will be approxiamtely normal

Question 10

for what values of n will the sample distribution be nearly normal?

Answer 10

For most distributions, 𝑛>25 will give a sampling distribution that is nearly normal

Question 11

how can you use the sample proportion to provide an estimate of population proportion?

Answer 11

The mean of the sample proportion is:
𝐸(𝑝̂ )=𝜇_𝑝̂ =𝑝
The standard deviation of the sample proportion is:
𝜎_𝑝̂ =√(𝑝(1−𝑝)/𝑛)
the standard normal approximation:
𝑍=(𝑝̂−𝑝)/𝜎_𝑝̂